[Physics] Are pressure and velocity uniform for a cross-section of fluid flow

bernoulli-equationfluid dynamics

Figure 1

The flow depicted in the figures is irrotational and the fluid is nonviscous, incompressible.

My questions –

  1. Is pressure uniform and constant for the whole cross-section(depicted by $A_1$ and $A_2$)?
  2. Is velocity of every particle constant for a cross-section perpendicular to the tube boundary or there exists a velocity gradient in the pipe perpendicular to the flow?

If the above written were true, can equation of continuity $A_1 v_1 = A_2 v_2$ be applied to finite cross-sectional areas? I don't think so.

Note – I initially asked this question but I feel that was very broad so this new question.

Best Answer

Given your assumptions, this flow satisfies bernoulli. From the continuity equation, you can see that velocity will vary as the cross sectional area varies. Assuming the vertical distance change is very minor, it can be dropped from the bernoulli equation. From the continuity equation, v2 is less than v1, so that p2 is greater than p1. For a given cross section which is perpendicular to the flow, the magnitude of the velocity will be the same, since there are no friction forces acting on the fluid.

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